A local council owns an underground railway

system and the railways managers have to make a decision on whether to lower fares

in an attempt to increase passenger numbers. If they decide to reduce fares

they will then have to decide whether to launch an advertising campaign on

local television to increase awareness of the fare reduction. If fares remain

the same then it is estimated that there is a 0.7 probability that the mean

number of passengers carried per day over the next year will equal 20 000 and a

0.3 probability that it will decline to 15 000. The annual profits associated

with

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A local council owns an underground railway

system and the railways managers have to make a decision on whether to lower fares

in an attempt to increase passenger numbers. If they decide to reduce fares

they will then have to decide whether to launch an advertising campaign on

local television to increase awareness of the fare reduction. If fares remain

the same then it is estimated that there is a 0.7 probability that the mean

number of passengers carried per day over the next year will equal 20 000 and a

0.3 probability that it will decline to 15 000. The annual profits associated

with these passenger numbers are estimated to be $3 million and $1 million, respectively.

If fares are reduced, but television advertising is not used, then it is

thought that there is a 0.6 probability that the mean number of passengers

carried will increase to 25 000 and a 0.4 probability that it will increase to

22 000. The resulting profits generated by these passenger numbers are

estimated to be $2 million and $1.7 million, respectively. Advertising the fare

reduction on television would increase the probability of an increase to a mean

of 25 000 passengers to 0.8 and reduce the probability that the mean will be 22

000 to 0.2. However, it would reduce the profits associated with these mean passenger

numbers by $0.6 million. The railways objectives are to maximize profit and to

maximize passenger numbers (since this brings environmental benefits such as

reduced traffic congestion).

(a) Utility functions for the mean numbers

of passengers carried and profit have been elicited from the railways chief

executive and these are given below. Plot these utility functions and interpret

them.

(b) The elicitation session revealed that,

for the chief executive, mean number of passengers and profit are mutually

utility independent. You are reminded that, in this case, a two-attribute utility

function can be obtained from:

u(x1, x2) = k1u(x1) + k2u(x2) +

k3u(x1)u(x2)

where k3 = 1 _ k1 _ k2.

The elicitation session also revealed that

k1 = 0.9 and k2 = 0.6, where attribute number 1 is the mean number of

passengers. Determine the policy that the railway should pursue in the light of

the above utilities and comment on your answer.

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